New Bound State Energies for Spherical Quantum Dots in Presence of a Confining Potential Model at Nano and Plank ’ s Scales

In present work, the exact analytical bound-state solutions of modified Schrödinger equation with Modified central potential consisting of a Cornellmodified plus pseudoharmonic harmonic potential (MCMpH) have been presented using both Boopp’s shift method and standard perturbation theory, we have also constructed the corresponding noncommutative Hamiltonian which containing two new terms, the first one is modified Zeeman effect and the second is new spin-orbital interaction. The theoretical results show that the automatically appearance for both spin-orbital interaction and modified Zeeman Effect leads to the degenerate to energy levels to ( ) 1 2 2 + l sub states.


Introduction
It is well known that to study any quantum chemical-physical model, in different fields of sciences like atomic, nuclear, molecular, harmonic and harmonic spectroscopy, we need to solve the non relativistic Schrödinger equation and relativistic two equations: Klein-Gordon and Dirac .To obtain profound interpretation in Nano and plank's scales, much work in case of the noncommutative space-phase at two, three and N generalized dimensions has been done for solving the three fundamental previously equations .The notions of noncommutativity of space and phase developed on based to the Seiberg-Witten map, Boopp's shift method and the star product, defined on the first order of two infinitesimal parameters antisymmetric as : ..… (1) Which allow us to obtaining the two new non nulls commutators and [ ] * j i p p , ˆ, respectively as: ………….…..…….. (2) It is important to notice, that the Boopp's shift method will be applying in this paper instead of solving the (NC-3D: RSP) with star product, the Schrödinger equation will be treated by using directly usual commutators on quantum mechanics, in addition to the two commutators [29][30][31][32][33][34][35][36][37][38][39][40][41][42][43]: rewritten the radial wave function ( ) r l n, ψ to the form [48]: , , 1 φ ψ = …………….………..….…..…….. (8) Then, the equation (7) reduces to following form [48]: Where, ( ) ( ) = ε and then, the complete normalized wave functions and corresponding energies for the ground state, the first existed states, and th n excited state, respectively [48]: and the two normalized constants ( l N 0 , l N 1 ) are given by [48]: and .

Noncommutative Phase-space Hamiltonian for (MCMpH)
Formalism of Boopp's shift Based on the previous works [31][32][33][34][35][36][37][38][39][40][41][42][43], we give a brief review to the fundamental principles of modified Schrödinger equation in (NC-3D: RSP), to achieve this goal we apply the important 4-steps on the ordinary (SE): The main goal of this work is to extend our study in reference [41] for the potential (MCMpH) including new term into noncommutative three dimensional spaces and phases on based to the principal reference [48] to discover the new spectrum and possibility to obtain new applications for the modified potential in different fields.The rest of present search is organized as follows: In next section, we give briefly review to the Schrödinger equation with (CMpH) in three dimensional spaces.In section 3, we shall briefly introduce the fundamental concepts of Boopp's shift method and then we apply this method to derive the deformed potential and noncommutative spin-orbital Hamiltonian.In section 4, we apply perturbation theory to find the spectrum for ground stat and first excited states and then we deduce the spectrum produced automatically by the external magnetic field.In section 5, we conclude the global noncommutative Hamiltonian and we resume the global spectrum for (MCMpH) in first order of two infinitesimal parameters' Θ and θ in (NC-3D: RSP).
Finally, the important found results and the conclusions are discussed in the last section.

…….. (19)
It is clear that, the first 4-terms in eq. ( 19) represent the ordinary potential while the rest term is produced by the deformation of space and phase.The global perturbative potential operators for studied potential (MCMpH) in both (NC-3D: RSP) will be written as:

E
for spin up and spin down, respectively, at first order of two parameters Θ andθ .In order to achieve this goal, we apply the standard perturbation theory: Where, a m 0 2 = β and 1 ' Where, the five terms ( ) are given by: In order to obtain the above integrals, we applying the .
Where the two factors and are given by: It's important to notice that the above two terms and are represent the noncommutative geometry of space and phase, respectively.

The Exact Spectrum of First Excited States
( 1) following special integration [50]:

New Bound State Energies for Spherical Quantum Dots in Presence of a Confining Potential Model at Nano and Plank's Scales
Maireche.
The aim of this subsection is to obtain the new modifications to the energy levels for first excited states and corresponding spin up and spin down, respectively at first order of two parameters Θ and θ for (MCMpH) which are obtained by applying the standard perturbation theory as: Where, ( ' ) and then a direct simplification to the above equations (32.1) and (32.2) gives: ............... (33) .............. (34) Where, the 15-terms are given by: In order to obtain the results of above equations, we apply the special integral which represents by eq. ( 28):       The explicit results obtained above allow us to get the exact modifications

for (MCMpH) in (NC: 3D-RSP)
On the other hand, it is possible to found another automatically symmetry for (CMpH) related to the influence of an external uniform magnetic field, generated from the effect of the new geometry of space and phase, it is deduced by the two following two replacements: Here χ and σ are infinitesimal real two proportional's constants and to simplify the calculations we choose the magnetic field k B B = and then we can make the following translation:  43.2) of eigenvalues of energies are reels and then the noncommutative diagonal Hamiltonian operator will be Hermitian operator.Furthermore, we can obtain the explicit physical form of this operator based on the results ( 23) and ( 42) for (CMpH), its represent by diagonal noncommutative matrix of order ( )

and and
As it's mentioned in our previous works [31][32][33][34][35][36][37][38][39][40][41][42][43], the atomic quantum number m can be taken ( and we have also two values for 2 1 ± = l j , thus every state in usually three dimensional space of (MCMpH) will be in (NC-3D: RSP): ( ) It is important to notice that the appearance of the polarization states of a fermionic particle indicates the validity of the results in the field of high energy where Dirac equation is applied, which allowing to the validity to results of present search on the Plank's and Nano scales level.If we make the limits ( ) ( ) we can obtain all the results of ordinary quantum mechanics.

Conclusion
In this article, we have investigated the solutions of the Schrödinger equation for modified (MCMpH) potential.We showed that the obtained degenerated spectrum for the modified studied potential depended by new discrete atomic quantum numbers: basis on quantum mechanics, then the operator

E
≡ ˆ denote to the ordinary operator of Hamiltonian for of Zeeman Effect in quantum mechanics.To obtain the exact noncommutative magnetic modifications of energy ( ) for (MCMpH) we just replaced the ....................(39) 3-parameters: + k , Θ and θ in the Eqs.(30.1) and (38.1) by the following new parameters: m with ( magnetic modifications of spectrum corresponding the ground states and first excited states, respectively.The new global exact spectrum of lowest excited states for (MCMpH) in (NC-3D: RSP) produced by the diagonal elements of noncommutative Hamiltonian operator .It is clearly, that the obtained previous results which are presented by Eqs.(30.1), (30.2), (38.1), (38.2), (43.1) and (

r 2 p
New

Bound State Energies for Spherical Quantum Dots in Presence of a Confining Potential Model at Nano and Plank's Scales Maireche.
Now, the aim of this subsection is to obtain the modifications to the energy levels for ground states ...............(37.3)

New Bound State Energies for Spherical Quantum Dots in Presence of a Confining Potential Model at Nano and Plank's Scales Maireche.
i L i = ......(37.4)